Identifier
Values
0 => 1 => ([(0,1)],2) => 0
1 => 1 => ([(0,1)],2) => 0
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
01 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
10 => 11 => ([(0,2),(2,1)],3) => 0
11 => 11 => ([(0,2),(2,1)],3) => 0
000 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
001 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
010 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
011 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
100 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
101 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
110 => 111 => ([(0,3),(2,1),(3,2)],4) => 0
111 => 111 => ([(0,3),(2,1),(3,2)],4) => 0
1110 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
1111 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
11110 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
11111 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
111110 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 0
111111 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 0
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Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Map
valleys-to-peaks
Description
Return the binary word with every valley replaced by a peak.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. This map replaces every valley with a peak.
Map
poset of factors
Description
The poset of factors of a binary word.
This is the partial order on the set of distinct factors of a binary word, such that $u < v$ if and only if $u$ is a factor of $v$.